where \(|i)\in \mathcal {H}\)  stands for the \(i\)  number states of photons with the normalization factor \(Z_K\) . We adopt Lloyd's quantum illumination protocol [1]} to formulate the quantum noise state as a coherent state represented by a large mean photon number. As a consequence, we can \(\rho _0,\rho _1\)  as the density matrices corresponding to the hypotheses \(H_0,H_1\)  as follows:
\(\begin{aligned}H_0:\rho _0 &= (1-N_B)|0\rangle \langle 0| + N_B{|k\rangle \langle k|},\\H_1:\rho _1 &= (1-x)\left((1-N_B)|0\rangle \langle 0| + N_B{|k\rangle \langle k|}\right) \\&\quad + x|l\rangle \langle l|,\end{aligned}\) 
